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The Schilder theorem gives a large deviation principle for small Brownian motion paths. For standard Brownian motion $W$ starting at zero on $[0,T]$, the laws of $\sqrt{\epsilon }W$ on continuous paths with the uniform norm have speed $1/\epsilon$ and good rate functionIn particular, the zero starting value and finite-energy path class are part of the rate. Typical Brownian motion paths are not themselves absolutely continuous; the rate describes the exponential cost of neighborhoods of prescribed paths.^[ref]

For paths constrained by $\varphi (T)=b$, the Cauchy-Schwarz inequality gives ${\int }_0^T|\dot{\varphi }{|}^{2}dt\ge b^2/T$, with equality for $\varphi (t)=bt/T$. Thus the least endpoint cost is $b^2/(2T)$, agreeing with the exponential scale of the normal distribution of $\sqrt{\epsilon }{W}_T$.

Vitalii Konarovskyi is a mathematician working in probability theory working on stochastic analysis and interacting particle systems. In his paper on coalescing heavy diffusion particles, he constructed a modified Arratia flow in which particles carry conserved mass and their diffusion rate is inversely proportional to that mass, starting from uniformly distributed mass on an interval.^[ref]

The mechanism is easy to picture: after particles merge, the heavier cluster diffuses more slowly. A rigorous construction must accommodate a continuum of initial particles, rather than only a finite collection of independent paths. His work develops a stochastic integral for the resulting flow and describes total local time through an occupation measure. His authored notes on the large deviation principle also explain how unlikely fluctuations acquire quantitative path costs.^[ref]